Legs and ratio

The legs of a right triangle are in the ratio a:b = 6:8. The hypotenuse has a length of 61 cm.
Calculate the perimeter and area of this triangle.

Final Answer:

p =  146.4 cm
S =  893.04 cm²

Step-by-step explanation:

$$\begin{gathered} a:b=6:8 \\ c=61\text{cm} \\ a:b=6:8=6x:8x \\ {c}^2=a^2+b^2=(6x{)}^2+(8x{)}^2 \\ x=\sqrt{\frac{c^2}{6^2+8^2}}=\sqrt{\frac{61^2}{6^2+8^2}}=\frac{61}{10}=6.1\text{cm} \\ a=6\cdot x=6\cdot 6.1=\frac{183}{5}=36.6\text{cm} \\ b=8\cdot x=8\cdot 6.1=\frac{244}{5}=48.8\text{cm} \\ p=a+b+c=36.6+48.8+61=146.4\text{cm} \end{gathered}$$

$S=\frac{a\cdot b}{2}=\frac{36.6\cdot 48.8}{2}=893.04{\text{cm}}^2$

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